Notes on the Construction of the Moduli Space of Curves

NOTES ON THE CONSTRUCTION OF THE MODULI SPACE OF CURVES

DAN EDIDIN

The purpose of these notes is to discuss the problem of moduli for curves of genus g ≥ 3 1 and outline the construction of the (coarse) moduli scheme of stable curves due to Gieseker. The notes are broken into 4 parts. In Section 1 we discuss the general problem of constructing a moduli “space” of curves. We will also state results about its properties, some of which will be discussed in the sequel. We begin Section 2 by recalling from [DM] (see also [Vi]) the definition of a groupoid, and define the moduli groupoid of curves, as well as the quotient groupoid of a scheme by a group. We then discuss morphisms and fiber products in the 2-category of groupoids. Once this is in place we state the geometric conditions required for a groupoid to be a stack, and prove that the quotient groupoid of a scheme by a group is a stack. After discussing properties of morphisms of stacks, we define a Deligne-Mumford stack and prove that if a group acts on a scheme so that the stabilizers of geometric points are finite and reduced then the quotient stack is Deligne-Mumford. In the last part of Section 2 we talk about some basic algebro-geometric properties of Deligne-Mumford stacks. In Section 3 the notion of a stable curve is introduced, and we define the groupoid of stable curves. The groupoid of smooth curves is a sub- groupoid. We then prove that the groupoid of stable curves of genus g ≥ 3 is equivalent to the quotient groupoid of a Hilbert scheme by the action of the projective linear group with finite, reduced stabilizers at geometric points. By the results of the previous section we can conclude the the groupoid of stable curves is a Deligne-Mumford stack defined over Spec Z (as is the groupoid of smooth curves). We also discuss the results of [DM] on the irreducibility of the moduli stack. We begin Section 4 by defining the moduli space of a Deligne-Mumford stack, proving that a geometric quotient of a scheme by a group is the The author was partially supported by an N.S.A grant during the preparation of this work. 1Because every curve of genus 1 and 2 has non-trivial automorphisms, the problem of moduli is more subtle in this case than for curves of higher genus 1 moduli space of the quotient stack. We then discuss the method of geometric invariant theory for constructing geometric quotients for actions of reductive groups. Finally, we briefly outline Gieseker’s approach to constructing the coarse moduli space over an algebraically closed field as the quotient of Hilbert schemes of pluricanonically embedded stable curves. Acknowledgments: These notes are based on lectures the author gave at the Weizmann Institute, Rehovot, Israel in July 1994. It is a pleasure to thank Amnon Yekutieli and Victor Vinnikov for the invitation, and for many discussions on the material in these notes. Special thanks to Angelo Vistoli for a careful reading and many critical comments. Thanks also to Alessio Corti, Andrew Kresch and Ravi Vakil for useful discussions.

1. Basics Definition 1.1. Let S be a scheme. By a smooth curve of genus g over S we mean a proper, smooth family C → S whose geometric fibers are smooth, connected 1-dimensional schemes of genus g. Remark 1.1. By the genus of a smooth, connected curve C over an 0 1 algebraically closed field, we mean dim H (C, ωC ) = dim H (C, OC ) = g, where ωC is the sheaf of regular 1-forms on C. If the ground field is C, then C is a smooth, compact Riemann surface, and the algebraic definition of genus is the same as the topological one. The basic problem of moduli is to classify curves of genus g. As a start, it is desirable to construct a space Mg whose geometric points represent all possible isomorphism classes of smooth curves. In the language of complex varieties, we are looking for a space that parametrizes all possible complex structures we can put on a fixed surface of genus g. However, as modern (post-Grothendieck) algebraic geometers we would like Mg to have further functorial properties. In particular given a scheme S, a curve C → S should correspond to a morphism of S to Mg (when S is the spectrum of an algebraically closed field this is exactly the condition of the previous paragraph). In the language of functors, we are trying to find a scheme Mg which represents the functor FMg : Schemes → Sets which assigns to a scheme S the set of isomorphism classes of smooth curves of genus g over S. Unfortunately, such a moduli space can not exist because some curves have non-trivial automorphisms. As a result, it is possible to construct non-trivial families C → B where each fiber has the same isomorphism class. Since the image of B under the corresponding map to the moduli 2 space is a point, if the moduli space represented the functor FMg then C → B would be isomorphic to the trivial product family. Given a curve X and a non-trivial (finite) group G of automorphisms of X we construct a non-constant family C → B where each fiber is isomorphic to X as follows. Let B0 be a scheme with a free G action, and let B = B0/G be the quotient. Let C0 = B0 × X. Then G acts on C0 by acting as it does on B0 on the first factor, and by automorphism on the second factor. The quotient C0/G is a family of curves over B. Each fiber is still isomorphic to X, but C will not in general be isomorphic to X × B. There is however, a coarse moduli scheme of smooth curves. By this we mean:

Deﬁnition 1.2. ([GIT, Deﬁnition 5.6, p.99]) There is a scheme Mg and a natural transformation of functors φ : FMg → Hom(·, Mg) such that

1. For any algebraically closed ﬁeld k, the map φ : FMg (Ω) → Hom(Ω, Mg) is a bijection, where Ω = Spec k.

2. Given any scheme M and a transformation ψ : FMg → Hom(·,M) there is a unique transformation χ : Hom(·, Mg) → Hom(·,M) such that ψ = χ ◦ φ. The existence of φ means that given a family of curves C → B there is an induced map to Mg. We do not require however, that a map to moduli gives a family of curves (as we have already seen a non-constant family with iso-trivial fibers). However, condition 1 says that giving a curve over an algebraically closed field is equivalent to giving a map of that field into Mg. Condition 2 is imposed so that the moduli space is a universal object. In his book on geometric invariant theory Mumford proved the following theorem. Theorem 1.1. ([GIT, Chapter 5]) Given an algebraically closed field k there is a coarse moduli scheme Mg of dimension 3g −3 defined over Spec k, which is quasi-projective and irreducible. The proof of this theorem will be subsumed in our general discussion of the construction of Mg, the moduli space of stable curves. A natural question to ask at this point, is whether Mg is complete (and thus projective). The answer is no. It is quite easy to construct curves C → Spec O with O a D.V.R. which has function field K, where C is smooth, but the fiber over the residue field is singular. Since C is smooth, the restriction CK → Spec K is a smooth curve over Spec K, so there is a map Spec K → Mg. The existence of such a family does 3 not prove anything, since we must show that we can not replace the special fiber by a smooth curve. The total space C˜ of a family with modified special fiber is birational to C. Since C and C˜ are surfaces (being curves over 1-dimensional rings) there must be a sequence of birational transformations (centered in the special fiber) taking one to the other. It appears therefore, that it suffices to construct a family C → Spec O such that no birational modification of C centered in the special fiber will make it smooth. Unfortunately, because Mg is only a coarse moduli scheme, the existence of such a family does not prove that Mg is incomplete. The reason is that there may be a map Spec O → Mg extending the original map Spec K → Mg without ˜ there being a family of smooth curves C → Spec O extending CK → Spec K. However, we will see when we discuss the valuative criterion of properness for Deligne-Mumford stacks that it suffices to show that for every finite extension K ⊂ K0 we can not complete the induced 0 family of smooth curves CK0 → Spec K to a family of smooth curves C0 → Spec O0, where O0 is the integral closure of O in K0.

Example 1.1. The following family shows that M3 is not complete. It can be easily generalized to higher genera. Consider the family x4 + xyz2 + y4 + t(z4 + z3x + z3y + z2y2) of quartics in P2 × Spec O where O is a D.V.R. with uniformizing parameter t. The total space of this family is smooth, but over the closed point the fiber is a quartic with a node at the point (0 : 0 : 1) ∈ P2. Moreover, even after base change, any blow-up centered at the singular point of the special fiber contains a (−2) curve, so there is no modification that gives us a family of smooth curves.

Since Mg is not complete, a natural question is to ask whether it is aﬃne. The answer again is no. This follows from the fact that Mg has a projective compactiﬁcation in the moduli space of abelian varieties, such that boundary has codimension 2. In particular, this means that there are complete curves in Mg. On the other hand, Diaz proved the following theorem ([Di]).

Theorem 1.2. Any complete subvariety of Mg has dimension less than g − 1. It is not known how close this bound is to being sharp. Finally we state a spectacular theorem proved by Harris-Mumford, Harris and Eisenbud-Harris.

Theorem 1.3. For g > 23, Mg is of general type. Remark 1.2. The importance of this theorem is that until its proof, there was some belief that Mg was rational, or at least unirational. 4 The reason is that for g = 3, 4, 5 it is very easy to show that Mg is unirational and not much more diﬃcult to show that is in fact rational. For g ≤ 10, the unirationality of Mg had been aﬃrmed by the Italian school. (For a nice discussion of the rationality of moduli spaces of curves of low genus, see Dolgachev’s article [Do].)

2. Stacks Let S be a scheme, and let S = (Sch/S) be the category of schemes over S. 2.1. Groupoids. Deﬁnition 2.1. A category over S is a category F together with a covariant functor pF : F → S. If B is an object of S we say X lies over B if pF (X) = B.

Definition 2.2. (see also [Vi, Definition 7.1]) If (F, pF ) is a category over S, then it is a groupoid over S if the following conditions hold: (1) If f : B0 → B is a morphism in S, and X is an object of F lying over B, then there is an object X0 over B0 and a morphism φ : X0 → X such that pF (φ) = f. (2) Let X,X0,X00 be objects of F lying over B,B0,B00 respectively. If φ : X0 → X and ψ : X00 → X are morphisms in F , and h : B0 → B00 is a morphism such that pF (ψ) ◦ h = pF (φ) then there is a unique 0 00 morphism χ : X → X such that ψ ◦ χ = φ and pF (χ) = h. Remark 2.1. Condition (2) implies that a morphism φ : X0 → X of objects over B0 and B respectively is an isomorphism if and only if 0 pF (φ): B → B is an isomorphism. (To see that pF (φ) being an isomorphism is sufficient to ensure that φ is an isomorphism, apply condition (2) where one of the maps is pF (φ) and the other the identity, −1 0 −1 0 and lift pF (φ) : B → B to φ : X → X . The other direction is trivial.) Define F (B) to be the subcategory consisting of all objects X such that pF (X) = B and morphisms f such pF (f) = idB. Then F (B) is a groupoid; i.e. a category where all morphisms are isomorphisms. This is the reason we say that F is a groupoid over S. Condition (2) also implies that the object X0 over B0 in condition (1) is unique up to canonical isomorphism. This object will be called the pull-back of X via f and denoted f ∗X. Moreover, if X →s X0 is a f ∗s morphism in F (B) then there is a canonical morphism f ∗X → f ∗X0. f In other words, given a morphism B0 → B of S-schemes, there is an induced functor f ∗ : F (B) → F (B0). Note that f ∗ is actually a covariant functor. 5 Example 2.1. If F : S → Sets is a contravariant functor, then we can associate a groupoid (also called F ) whose objects are pairs (B, β) where B is an object of S and β ∈ F (B). A morphism (B0, β0) → (B, β) is an S-morphism f : B0 → B such that F (f)(β) = β0. In this case F (B) in the groupoid sense is just the set F (B) in the functor sense; i.e. all morphisms in the groupoid F (B) are the identities. In particular, if X is any S-scheme then its functor of points gives φ a groupoid X. Objects are X-schemes, i.e. morphisms B → X, and a φ0 φ f morphism from B0 → X0 to B → X is a morphism B0 → B0 such that 0 ψf = ψ . The functor pX simply forgets the X-structure, and views schemes and morphisms as being over S. Example 2.2. If X/S is a scheme and G/S is a flat group scheme (of finite type) acting on the left on X then we define the quotient groupoid [X/G] as follows. The sections (i.e. objects) of [X/G] over B are G- principal bundles E → B together with a G-equivariant map f : E → X. A morphism from E0 → B0 with equivariant map f 0 : E0 → B0 to E → B is a cartesian diagram g E0 → E ↓ ↓ B0 → B where g is an equivariant map such that gf = f 0. If the action is free and a quotient scheme X/G exists, then there is an equivalence of categories between [X/G] and the groupoid X/G. Example 2.3. The central example in these notes is the moduli groupoid

FMg deﬁned over Spec Z. The objects of FMg are smooth curves as deﬁned in part 1. A morphism from X0 → B0 to X → B is a cartesian diagram X0 → X ↓ ↓ B0 → B

The functor FMg → Sch Z sends X → B to B. We will eventually prove that FMg is a quotient groupoid as in the previous example.

A related groupoid is the universal curve FCg , also deﬁned over

Spec Z. Objects of FCg are smooth curves X → B together with a section σ : B → X. A morphism is a cartesian diagram which is compatible with the sections. Remark 2.2 (Warning). The groupoid we have just deﬁned is not the groupoid associated to the moduli functor we deﬁned in Part 1. The groupoid here is not a functor, since if X/B is a curve with non-trivial 6 automorphisms F (B) will not be a set because there are morphisms which are not identities. (A set is a groupoid where all the morphisms are identities.) 2.2. Morphisms of groupoids.

Deﬁnition 2.3. (a) If (F1, pF1 ) and (F2, pF2 ) are groupoids over S then a morphism F1 → F2 is a functor p : F1 → F2 such that ppF2 = pF1 . (b) A morphism p is called an isomorphism if it is an equivalence of categories.

Example 2.4. The functor FCg → FMg defined by forgetting the section is a morphism of groupoids. Here are some more subtle examples: Example 2.5. If f : X → Y is an morphism of schemes then it induces p a morphism of groupoids X → Y in a fairly obvious way: Objects of X are X-schemes, which, via the morphism f, can be viewed as Y - fu schemes, i.e. objects of Y . Thus, p(B →u X) = B → Y . If s : B → B0 is a morphism of X-schemes, then p(s) is the morphism s viewed now as a morphism of Y -schemes. Conversely, if p : X → Y is a morphism of groupoids over S then, f because p preserves the projection to S, p(X →id X) = X → Y for some morphism f. Yoneda’s lemma implies that p is induced by the morphism of schemes f : X → Y . More generally, if F is a groupoid and B is a scheme then giving a p morphism B → F is equivalent to giving an object X in F (B). The object X is simply p(B →id B). Example 2.6. We can view the category S of S-schemes, (which is trivially a groupoid over S) as the groupoid S. If F is a groupoid over S then functor pF is then a morphism of groupoids F → S. Example 2.7. Let X/S be a scheme and G/S a group scheme acting on the left on X. Then we can define a morphism p : X → [X/G] as follows: If B →s X is an object of X(B) then p(s) is the bundle G × B → B where G acts by left translation on G and trivially on B. The equivariant map G×B → X is given by the formula (g, b) 7→ gs(b). If f : B0 → B is a morphism in X then p(f) is the cartesian diagram id×f G × B0 → G × B ↓ ↓ f B0 → B 7 Remark 2.3. Because isomorphisms of stacks are defined as equivalences of categories some subtleties arise. In particular an isomorphism p : F1 → F2 need not have an inverse; i.e. there need not be a functor q : F2 → F1 such that pq = idF2 and qp = idF1 . However, any equivalence of categories has a quasi-inverse; that is a functor q : F2 → F1 such that pq (resp. qp) are naturally isomorphic to the identity functors 1F2 (resp. 1F2 ). Moreover, if p : F2 → F1 is a equivalence which is a morphism of S-groupoids, then then there is a quasi-inverse which is a morphism of groupoids. The usual way to describe this point is to say that groupoids over S form a 2-category. This 2-category has objects, which are the groupoids; 1-morphisms, which are functors, and 2-morphisms, which are natural isomorphisms of functors. In other words, the category of groupoids contains extra information about isomorphisms between morphisms. For the most part, the fact that groupoids (and thus stacks) form a 2-category will not require too much thought but there a few situations where it is relevant. For example, as we discuss below, a cartesian diagram commutes only up to homotopy. A more geometric situation is the valuative criterion, where (see Theorem 2.2) we require two extensions of a morpisms be isomorphic. The following proposition shows that the notion of isomorphism of groupoids is an extension of our notion of isomorphism of schemes. Proposition 2.1. Let X and Y be S-schemes. Then there is an iso- f morphism X → Y as S-schemes if and only if there is an equivalence p of S groupoids X → Y

p Proof. If f is an isomorphism, then the induced functor X → Y is in q fact a strong equivalence; i.e. the functor Y → X induced by f −1 has the property that that pq = idY and qp = idX . p Conversely, suppose that X → Y is an isomorphism of groupoids q with quasi-inverse Y → X. As we saw in Example 2.5 the functor p (resp. q) is induced by a morphism f : X → Y (resp. g : Y → X). gf Then qp(X →id X) = X → X. Since q and p are equivalences X →id X gf and X → X are isomorphic as X-schemes. Hence gf : X → X is an automorphism. Likewise, fg : Y → Y is also an automorphism. Therefore, f : X → Y must be an isomorphism. Remark 2.4. If B is a scheme then from now on we will use the simpler notation B → F (resp. F → B) to refer to a morphism B → F (resp. F → B). 8 2.3. Fiber products and cartesian diagrams. Let F and G be groupoids over S. If f : F → G and h : H → G are morphisms of groupoids, then we define the fiber product F ×G H to be the following S-groupoid. Objects are triples (x, y, ψ) where (x, y) ∈ F (B) × H(B) and ψ : g(x) → h(y) is an isomorphism in G(B). Here B is a fixed scheme in S. Suppose (x0, y0, ψ0) is another object with (x0, y0) ∈ F (B0) and ψ0 : g(x) → h(y) an isomorphism in G(B0). A morphism from (x0, y0, ψ0) β to (x, y, ψ) is a pair of morphisms x0 →α x, y0 → y lying over the same morphism B0 → B, such that ψ ◦ f(α) = g(β) ◦ ψ0.

By construction there are obvious functors p : F ×G H → F and q : F ×G H → H. Note however, that the diagram

F ×G H → H ↓ ↓ F → G does not commute, since fp(x, y, ψ) = f(x) and gq(x, y, ψ) = g(y). The objects f(x) are g(y) isomorphic but not necessarily equal. There is however a natural isomorphism between the functors fp and gq. Following the language of 2-categories we say that such a diagram is 2-commutative. More generally given a 2-commutative diagram of groupoids T → H ↓ ↓ F → G there is a morphism T → F ×G H which is unique up to canonical isomorphism. If this morphism is an isomorphism then we say the diagram is cartesian.

Example 2.8. If X,Y,Z are schemes then X×Z Y is isomorphic X ×Y Z, so our notion of ﬁber product is an extension of the usual one for schemes. Remark 2.5. Despite the subtleties, this notion is correct for consider- ing base change. In particular, suppose that F is an S-groupoid and T → S a morphism of schemes. Then if B → T is a T -scheme, one can check that the groupoids F (B) and (F ×S T )(B) are equivalent; i.e. F ×S T is the T -groupoid obtained by base change to T .

2.4. Definition of a stack. Let (F, pF ) be an S-groupoid. Let B be an S-scheme and let X and Y be any objects in F (B). Define a contravariant functor IsoB(X,Y ) : (Sch /B) → (Sets) by associ- ating to any morphism f : B0 → B, the set of isomorphisms in 9 f g F (B0) between f ∗X and f ∗Y . Let B0 → B and B00 → B be B- schemes If h : B00 → B0 is a morphism of B-schemes (ie. g = fh) then by construction of the pullback, there are canonical isomorphisms ∗ ∗ ∗ ∗ ∗ ∗ ψX : g X → h f X and ψY : g Y → h f Y . We define a map 0 00 ∗ φ ∗ IsoB(X,Y )(B ) → IsoB(X,Y )(B ) as follows: If f X → f Y is an isomorphism then (since h∗ : F (B0) → F (B00) is a functor) we obtain an ∗ ∗ ∗ h φ ∗ ∗ −1 ∗ isomorphism h f X → h f Y . The composite, ψY ◦ h φ ◦ φX is the 00 image of φ in IsoB(X,Y )(B ). 0 If X = Y then IsoB(X,X) is the functor whose sections over B mapping to B are the automorphisms of the pull-back of X to B0. In the case of curves of genus g ≥ 2, Deligne and Mumford proved that IsoB(X,Y ) is represented by a scheme IsoB(X,Y ), because X/B and Y/B have canonical polarizations ([DM, p.84]). When X = Y then Deligne and Mumford prove directly that the IsoB(X,X) is finite and unramified over B ([DM, Theorem 1.11]). Applying the theorem to B = Spec k, where k is an algebraically closed field, this theorem proves that every curve has a finite automorphism group. The scheme IsoB(X,X) is naturally a group scheme over B. How- ever, in general it will not be flat over B. For example, if X/B is a family of curves, the number of points in the fibers of IsoB(X,X) over B will jump over the points b ∈ B where the fiber Xb has non-trivial automorphisms.

Definition 2.4. A groupoid (F, pF ) over S is a stack if (1) IsoB(X,Y ) is a sheaf in the étale topology for all B, X and Y . (2) If {Bi → B} is a covering of B in the étaletopology, and Xi is a collection of objects in F (Bi) with isomorphisms

φij : Xj|Bi×B Bj → Xi|Bi×B Bj in F (Bi ×B Bj) satisfying the cocycle condition. Then there is an object ' X ∈ F (X) with isomorphisms X|Bi → Xi inducing the isomorphisms φij above. Note that F is a groupoid associated to a functor then conditions (1) and (2) just assert that the functor is a sheaf in the étale topology. A functor represented by a scheme will be a stack, since condition (1) is trivially satisfied and (2) is equivalent to saying that the functor of points is a sheaf in the étale topology. The moduli functor we defined in Part 1 is not a stack, since it doesn’t satisfy condition (2). In particular, as noted above, given a curve C with automorphism group G and B0 → B a Galois cover with group G, there are two ways to descend 10 the family C × B0/B0 to a family over B, so a section of F over B, is not determined by its pull-back to an étalecover. However, the moduli groupoid defined above is a stack. We will not prove this here, but instead we will prove that the moduli groupoid is the quotient groupoid of a scheme by PGL(N + 1). Proposition 2.2. (cf. [Vi, Example 7.17]) Let G/S be a smooth2 affine group scheme over X, then the groupoid [X/G] defined above is a stack. Proof. Let e, e0 be sections of [X/G](B) corresponding to principal bundles E → B and E0 → B and G-maps f : E → X and f 0 : 0 0 E → X. Then IsoB(e, e ) is the étale sheaf which is the quotient of 0 (X ×X×X E ×B E ) by the free product action of G. Moreover, descent theory show that this sheaf is in fact a scheme. When E = E0 and f = f 0 then the isomorphisms correspond to elements of G which preserve f. In other words IsoB(e, e) is the is the stabilizer of the G−map f : E → X (see [GIT, Definition 0.4] for the definition of stabilizer). Since any principal bundle E → B is locally trivial in the étale pi topology it determines descent data as follows: Let {Bi → B} be an étalecover on which E → B is trivial. Then we have and equivariant ∗ isomorphisms φi : pi E → G×Bi. If φij is the pullback of φi to Bi×B Bj, then the φij’s satisfy the cocycle condition; i.e. φijφik = φjk. Descent theory gives us the opposite direction. Given principal bundles (not necessarily trivial) Ei → Bi and isomorphisms of Ei|Bi×B Bj →

Ej|Bi×B Bj satisfying the cocycle condition, there is a principal bundle ∗ E → B such that Ei = pi E. This is condition (2) in the definition of a stack. Example 2.9. If F, G, H are stacks over S then one can check that the fiber product F ×G H is also a stack. 2.5. Representable morphisms. Most of the material in this section is taken from [DM, Section 4]. Definition 2.5. A morphism f : F → G of stacks is said to be representable if for any map of a scheme B → G the fiber product F ×G B is isomorphic to a stack associated to a scheme. Remark 2.6. The definition of representable morphism is the one given by Deligne-Mumford in [DM]. Artin, who considered a larger category of stacks, extended the definition to require that if B → F is a map of an algebraic space (which we define below) then the the fiber product

2In fact the proposition holds for flat affine group schemes as well. 11 F ×G B is also an algebraic space. For the stacks we consider here, the Deligne-Mumford definition suffices. Example 2.10. Let G/S be a smooth affine group scheme acting on X. We saw that there is a projection morphism X → [X/G] which associates to any X-scheme Z →s X, the principal bundle G × Z with the equivariant map given by (g, z) 7→ gs(z). Suppose B → [X/G] is a morphism from a scheme corresponding to a principal bundle E → B with equivariant map E → X. Let us a consider the category B ×[X/G] X. Its objects are triples {B1,B2, ψ} where B1 → B, B2 → X and ψ is an isomorphism between the images in [X/G]. Since B1 and B2 lie over the same object in the base category, there is a scheme 0 0 0 f B such that B1 = B2 = B . If B → B is a morphism then the image of B0 is the principal bundle f ∗E → B0 with equivariant map ∗ 0 f 0 s f E → E → X. Thus objects of B ×[X/G] X are triples (B → B,B → X, ψ) where ψ : G × B0 →∼ f ∗E is a trivialization. The section b ∈ 0 0 0 ∗ B 7→ (1G, b ) gives a morphism B → f E → E. Conversely, suppose we are given a morphism B0 → E. Let f : B0 → E → B be the composite morphism. The diagonal gives is a canonical trivialization ∗ 0 of the bundle E ×B E → E and hence of f E → B . Thus we can factor the morphism B0 → E as B0 → f ∗E → E. A similar analysis shows that morphisms in B ×[X/G] X correspond to morphisms of E- schemes, so B ×[X/G] X is isomorphic to E. Therefore, the morphism X → [X/G] is representable.

Example 2.11. The morphism FCg → FMg forgetting the section is also representable. If B → FMg corresponds to a smooth curve C → B then B ×FMg FCg is represented by the scheme C. Let P be a property of morphisms of schemes which is stable under base change. Most properties of morphisms of schemes satisfy this property. For example, finite type, separated, proper, affine, flat, smooth (hence étale), etc. Definition 2.6. ([DM, Definition 4.3] A representable morphism of stacks f : F → G has property P if for all maps of scheme B → G the corresponding morphism of schemes F ×G B → B has property P. Example 2.12. The projection morphism X → [X/G] is smooth since for any B → [X/G] the corresponding map E → B is smooth because E is a principal bundle over E. 2.6. Definition of a Deligne-Mumford stack. Definition 2.7. A stack is Deligne-Mumford if 12 (1) The diagonal ∆F : F → F ×S F is representable, quasi-compact and separated. If the diagonal is proper, then we say that the stack is separated. (2) There is a scheme U and an étale surjective morphism U → F . Such a morphism U → F is called an (étale) atlas. Remark 2.7. The representability of the diagonal automatically implies that the functor IsoB(X,Y ) is representable. The reason is that if X and Y are objects in F (B), then IsoB(X,Y ) is represented by the fiber product (B × B)F ×F F where the map B × B → F × F is the product of the the maps B → F corresponding to the objects X and Y . Remark 2.8. In [DM], such a stack is called an algebraic stack. To con- form to current terminology we use the term Deligne-Mumford stack. A more general class of stacks was studied by Artin, and they are now called Artin stacks. The basic difference is that an Artin stack need only have a smooth atlas. An algebraic space is defined as an étalesheaf with an étale cover by a scheme. This the same as Deligne-Mumford stack where the diagonal is an embedding. An algebraic space is separated if the diagonal is a closed embedding. Remark 2.9. Condition (1) above is equivalent to the following condition:

(1’) Every morphism B → F from a scheme is representable, so condition (2) makes sense. (This fact is stated in [DM] and proved in [Vi, Prop 7.13]). Condition (2) asserts the existence of a universal deformation space for deformations over Artin rings. Remark 2.10 (Separated morphisms). The proof of the following lemma can be found in [L-MB]. Lemma 2.1. Let f : F → G be a morphism of stacks satisfying property (1) above. Then the diagonal ∆F/G : F → F ×G F is representable. As a consequence of the lemma we make the following extension of the notion of separated morphism of schemes. Deﬁnition 2.8. A morphism of Deligne-Mumford stacks is separated if ∆F/G is proper.

Vistoli also proves the following proposition: Proposition 2.3. [Vi, Prop 7.15] The diagonal of a Deligne-Mumford stack is unramified 13 As a consequence of this proposition we can prove [Vi, p. 666] Corollary 2.1. If F is a Deligne-Mumford stack, B quasi-compact, and X ∈ F (B) then X has only finitely many automorphisms. Remark 2.11. The are Artin stacks which are not Deligne-Mumford where each object has a finite automorphism group. In this case the diagonal is quasi-finite but ramified. Objects in the groupoid have infinitesimal automorphisms. This phenomenon only occurs in characteristic p, because all group schemes of finite type over a field of characteristic 0 are smooth.

Proof. Let B → F be map corresponding to X, and let B → F ×S F be the composition with diagonal. The pullback B ×F ×S F F can be identified with scheme IsoB(X,X). Since F is a Deligne-Mumford stack the map IsoB(X,X) is unramified over X. Furthermore, since B is quasi-compact, the map IsoB(X,X) → X can have only finitely many sections. Therefore, X has only finitely many automorphisms over B. The following theorem is stated (but not proved) in [DM, Theorem 4.21]. We give the proof below with a slight additional assumption. Theorem 2.1. Let F be a stack over a Noetherian scheme S. Assume that (1) The diagonal is representable, quasi-compact, separated and unramified, (2) There exists a scheme U of finite type over S and a smooth surjective S-morphism U → F , Then F is a Deligne-Mumford stack. Remark 2.12. This theorem says that condition (1) and the existence of a versal deformation space (condition (2)) is actually equivalent to the existence of a universal deformation space. Remark 2.13. We give the proof below under the additional assumption that the residue fields of the closed points of S are perfect. In particular we prove the theorem for stacks of finite type over Spec Z. Using the theorem we will prove that the stack of stable curves is a Deligne-Mumford stack of finite type over Spec Z. By assumption U is of finite type over S, so so it is relatively straightforward to reduce to the case that U is actually of finite type over Spec Z. Thus, the general statement can be reduced to the case we prove. However, we do not give the details here. Proof. The only thing to prove is that F has an étale atlas of finite type −1 over S. Let u ∈ U be any closed point. Set Uu = δ (u×S u) = u×F U. 14 Let z ∈ Uu be a closed point which is separable (i.e. étale) over u (The set of such closed points is dense in a smooth variety). Since Uu is smooth, the point z is cut out by a regular sequence in the local ring of Uu at z. The diagonal δ : F → F ×S F is unramified. Thus, the map Uu → u×S U obtained by pulling back the morphisms u×S U → F ×S F along the diagonal is unramified. We assume U is of finite type and that the residue fields of S are perfect. Thus, k(u) is a finite, hence separable, extension of the residue field of its image in S. Hence the morphism u×S U → U is unramified and so is the composition Uu → u×S U → U. Let x be the image of z in U. By [EGA4, 18.4.8] there are étale neigh- borhoods W 0 and U 0 of x and z respectively and a closed immersion W 0 ,→ U 0 such that the diagram commutes z0 ∈ W 0 ,→ U 0 étale ↓ ↓ étale z ∈ Uu → x ∈ U 0 Let z be any point lying over z. Let Zu be the closed subscheme 0 0 0 0 of U defined by lifts to OU of the local equations for z ∈ W . By 0 0 construction, Zu intersects U transversally at z . We will show that the induced morphism Zu → F is étale in a neighborhood of z. By definition, this means that for every map of a scheme B → F , the induced map of schemes B ×F Zu → B is étale in a neighborhood 0 of z ×F Zu. Since U → F is smooth and surjective, it it suffices to check that the morphism is étale after base change to U. 0 By construction, Zu ⊂ U is cut out by a regular sequence in a neighborhood of z0 ∈ U 0 (since z0 is a smooth point of W 0). Thus 0 0 Zu×F U → U ×F U is a regular embedding in a neighborhood of z ×F U. 0 0 Since U ×F U → U is smooth, we can apply [EGA4, Theorem 17.12.1], 0 0 0 and conclude that Zu ×F U → U is smooth in a neighborhood of z . Moreover, the relative dimension of this morphism is 0. Therefore, 0 Zu → F is étale in a neighborhood of z . Since U is of finite type over S, the Zu’s are as well. The union of the Zu’s cover F (since their pullbacks via the morphism U → F cover U). Also, U is Noetherian because it is of finite type over a Noetherian scheme. Thus a finite number of the Zu’s will cover the F . (To see this, we can pullback via the map U → F . The pullback of the Zu’s form an étale cover of U which is Noetherian.) The theorem has a useful corollary. Corollary 2.2. Let X/S be a Noetherian scheme of finite type and let G/S be a smooth affine group scheme (also of finite type over S) 15 acting on X such that the stabilizers of geometric points are finite and reduced. (i) [X/G] is a Deligne-Mumford stack. If the stabilizers are trivial, then [X/G] is an algebraic space.

(ii) The stack is separated if and only if the action is proper.

Proof. The condition on the action ensures that IsoB(E,E) is unramified over E for any map B → [X/G] corresponding to the principal bundle E → B. This in turn implies that the diagonal is also unramified, so condition (1) is satisfied. Furthermore, condition (2) is satisfied by the smooth map X → [X/G]. Suppose that [X/G] is separated, i.e. the diagonal [X/G] → [X/G]× [X/G] is proper and representable. One can check that

[X/G] ×[X/G]×[X/G] X × X is represented by the scheme G × X where the projection

[X/G] ×[X/G]×[X/G] X × X → X × X corresponds to the action map G×X → X ×X, so the action is proper. Conversely, suppose that map G × X → X × X is proper. This implies that the diagonal is proper after base change to X × X; i.e.,

[X/G] ×[X/G]×[X/G] X × X → X × X is proper. Let Z → [X/G] × [X/G] be any scheme and set W = [X/G] ×[X/G]×[X/G] [X/G]. We will use descent to show that W → Z is proper. 0 0 0 If Z = Z ×[X/G]×[X/G] X × X then the map W = W ×Z Z → Z is proper. Since X → [X/G] is smooth and surjective (in particular it is faithfully flat) Z0 → Z is as well. Descent theory for faithfully flat morphisms, implies that the map W → Z is proper. Therefore [X/G] is separated. Example 2.13. In order for a group action to be proper it must have have finite stabilizers. However, it is not difficult to construct examples of group actions which, despite having finite stabilizers, are not proper. In [GIT, Example 0.4] there is an example an SL(2, C) action on a 4- dimensional variety X which has trivial stabilizers but is not proper. The quotient [X/ SL(2, C)] is a non-separated algebraic space. 2.7. Further properties of Deligne-Mumford stacks. From now on the term stack will mean Deligne-Mumford stack, though we will often use the term Deligne-Mumford stack for emphasis. 16 Not all morphisms of stacks are representable, so we can not define algebro-geometric properties of these morphisms as we did for representable morphisms. However, if we consider morphisms of Deligne- Mumford stacks then we can define properties of morphisms as follows (see [DM, p. 100]). Let P be a property of morphisms of schemes which at source and target is of a local nature for the étale topology. This means that for any family of commutative squares

gi Xi → X fi ↓ f ↓ hi Yi → Y where the gi (resp hi) are étale and cover X (resp. Y ), then f has property P if and only if fi has property P for all i. Examples of such properties are f flat, smooth, étale, unramified, locally of finite type, locally of finite presentation, etc. Then if f : F → G is any morphism of Deligne-Mumford stacks we say that f has property P if there are étale atlases U → F , U 0 → G and a compatible morphism U → U 0 with property P. Likewise, if P is property of schemes which is local in the étale topology (for example regular, normal, locally Noetherian, of characteristic p, reduced, Cohen-Macaulay, etc.) then a Deligne-Mumford stack F has property P if for one (and hence every) étaleatlas U → F , the scheme U has property P. A stack F is quasi-compact if it has an étale atlas which is quasi- compact. A morphism f : F → G of stacks is quasi-compact if for any map of scheme, X → G the fiber product X ×G F is a quasi-compact stack. We can now talk about morphisms of finite type; a morphism of finite type is a quasi-compact morphism which is locally of finite type. Similarly, a stack is Noetherian if it quasi-compact and locally Noetherian. Definition 2.9. [DM, Definition 4.11] A morphism f : F → G is proper if it is separated, of finite type and locally over F there is a Deligne-Mumford stack H → F and a (representable) proper map H → G commuting with the projection to F and the original map F → G. H ↓ & F → G Remark 2.14. By a theorem of Vistoli [Vi, Prop. 2,6] and Laumon– Moret-Baily [L-MB, Theorem 10.1] every Noetherian stack has a finite 17 cover by a scheme. Using this fact we can say that a morphism F → G is proper if there is a finite cover X → F by a scheme such that the composition X → F → G is a proper representable morphism. (Recall that any morphism from a scheme to a stack is representable). Similarly we say that a morphism f : F → G of Noetherian stacks is (quasi)- finite if for any finite cover X → F , the composition X → F → G is representable and (quasi)-finite.

As is the case with schemes, there are valuative criteria for separation and properness ([DM, Theorem 4.18-4.19]). The valuative criterion for separation is equivalent to the criterion for schemes, but we only construct an isomorphism between two extensions.

Theorem 2.2. A morphism f : F → G is separated iﬀ the following condition holds: For any complete discrete valuation ring V and fraction ﬁeld K and any morphism f : Spec V → G with lifts g1, g2 : Spec V → F which are isomorphic when restricted to Spec K, then the isomorphism can be extended to an isomorphism between g1 and g2.

Theorem 2.3. A separated morphism f : F → G is proper if and only if for any complete discrete valuation ring V with ﬁeld of fractions K and any map Spec V → G which lifts over Spec K to a map to F , there is a ﬁnite separable extension K0 of K such that the lift extends to all of Spec V 0 where V 0 is the integral closure of V in K0

Remark 2.15. When F is a scheme it is easy to show to that there is a lift Spec V 0 → F if and only if there is a lift Spec V → F .

Example 2.14 (Angelo Vistoli). Here is an example showing the neces- sity of passing to a cover. Let G = {±1} act on X = A1 by by left multiplication and set C Y = A1. The double cover f : X → Y given by z 7→ z2 is clearly G-invariant (and in fact is a geometric quotient in the sense of [GIT] -see Section 4.1). In particular if E → B is a principal G-bundle with equivariant map to X, then composition E → X → Y is G-invariant. Hence, since E → B is a categorical quotient in the sense of [GIT] and there is a unique map B → Y making the diagram

E → X ↓ ↓ B → Y 18 commute. Moreover, E0 → E ↓ ↓ B0 → B is a cartesian diagram corresponding to a morphism in [X/G], then the morphism B0 → B is actually a morphism of Y -schemes. and we obtain a morphism ψ :[X/G] → Y . p It is easy to check that the double cover f : X → Y factors as X → ψ [X/G] → Y where p : X → [X/G] is the map defined in Example 2.7. Since G is finite, p is a finite surjective (and representable) morphism, and since f is finite, it follows that the non-representable morphism [X/G] → Y = A1 is finite. We wish to test the valuative criterion for this morphism. Let R = C[[t]] be the complete local ring of Y at 0, and let Spec R → Y be the obvious morphism. The restriction of the double cover X → Y to the generic point is a non-trivial degree 2 Galois covering corresponding the extension C(t) ⊂ C(u) where t = u2. We can view this cover as a principal G-bundle, giving us a lift Spec C(t) → [X/G]. How- ever, this lift can not be extended to all of Spec R since it has no non-trivial Galois covers. To obtain a lift we must first trivialize the bundle Spec C(u) → Spec C(t) by normalizing R in C(u). 2.8. The topology of stacks. Most of the topological properties of schemes make sense for Deligne-Mumford stacks. Thus in many ways we can think of them as spaces.

Remark 2.16 (Connectedness). If F1 and F2 are stacks over S, define ` the disjoint sum F = F1 F2 as follows: Objects are disjoint unions ` X1 X2 where X1 is an object of F1 and X2 is an object of X2.A 0 ` 0 ` morphism from X1 X2 → X1 X2 is specified by giving morphisms 0 0 X1 → X1 X2 → X2. A stack is connected if it is not the disjoint sum of two non-void stacks. Proposition 2.4. [DM, Proposition 4.14] A Noetherian stack F over a field is connected if and only if there is a surjective morphism X → F from a connected scheme. Remark 2.17 (Open and closed substacks). An open substack F ⊂ G is a full subcategory of G such that for any x ∈ Obj(F ), all objects in G isomorphic to x are also in F . Furthermore, the inclusion morphism F → G is represented by open immersions . In a similar way we can talk about closed (or locally closed) substacks. 19 In particular a stack is irreducible if it is not the disjoint union of two closed substacks. A normal stack is irreducible if and only if it is connected. (Since a being normal is an étalelocal property of schemes, a stack is normal iff it has a normal étaleatlas).

The following theorem is crucial to the proof of irreducibility of FMg in arbitrary characteristic. Theorem 2.4. [DM, Theorem 4.17(iii)] Let f : F → S be a proper flat morphism with geometrically normal fibers. Then the number of connected components of the geometric fibers is constant.

3. Stable curves In this section we discuss stable curves and the compactification of the moduli of curves to the moduli of stable curves. Definition 3.1. [DM, Definition 1.1] A Deligne-Mumford stable (resp. semi-stable) curve of genus g over a scheme S is a proper flat family C → S whose geometric fibers are reduced, connected, 1-dimensional schemes Cs such that: (1) Cs has only ordinary double points as singularities. (2) If E is a non-singular rational component of C, then E meets the other components of Cs in more than 2 points (resp. in at least 2 points). 1 (3) Cs has arithmetic genus g; i.e. dim H (OCs ) = g. Remark 3.1. Clearly, a smooth curve of genus g is stable. Condition (2) ensures that stable curves have finite automorphism groups, so that we will be able to form a Deligne-Mumford stack out of the category of stable curves. We will not use the notion of semi-stable curves until we discuss geometric invariant theory in Section 4.

Denote by FMg the groupoid over Spec Z whose sections over a scheme B are families of stable curves X → B. As is the case with smooth curves, we define a morphism from X0 → B0 to X → B as a cartesian diagram X0 → X ↓ ↓ . B0 → B 3.1. The stack of stable curves is a Deligne-Mumford stack. Let π : C → S be a stable curve. Since π is flat and its geometric fibers are local complete intersections, the morphism is a local complete intersection morphism. It follows from the theory of duality that there 20 is a canonical invertible dualizing sheaf ωC/S on C. If C/S is smooth, then this sheaf is the relative cotangent bundle. The key fact we need about this sheaf is a theorem of Deligne and Mumford [DM, p. 78]. Theorem 3.1. Let C →π S be a stable curve of genus g ≥ 2. Then ⊗n ⊗n ωC/S is relatively very ample for n ≥ 3, and π∗(ωC/S) is locally free of rank (2n − 1)(g − 1). Remark 3.2. When π is smooth, the theorem follows from the classical Riemann-Roch theorem for curves. The general case is proved by analyzing the locally free sheaf obtained by restricting ωC/S to the geometric fibers of C/S. In particular, if S = Spec k, with k alge- 0 braically closed, then ωC/S can be described as follows. Let f : C → C be the normalization of C (note C0 need not be connected). Let 0 x1, . . . , xn, y1, . . . , yn be the points of C such that the zi = f(xi) = f(yi) are the double points of C. Then ωC/S can be identified with the sheaf of 1-forms η on C0 regular except for simple poles at the x’s and y’s and with Resxi (η) + Resyi (η) = 0. As a result, if N = (2n − 1)(g − 1) − 1, then every stable curve can N be realized as a curve in P with Hilbert polynomial Pg,n(t) = (2nt − Pg,n 1)(g − 1). There is a subscheme (defined over Spec Z) Hg,n ⊂ HilbPN of the Hilbert scheme corresponding to n−canonically embedded stable curves. Likewise there is a subscheme Hg,n ⊂ Hg,n corresponding to n−canonically embedded smooth curves. A map S → Hg,n corresponds π to a stable curve C → S of genus g and an isomorphism of P(π∗(ωC/S)) with PN × S. Now, PGL(N + 1) naturally acts on Hg,n and Hg,n.

Theorem 3.2. FMg ' [Hg,n/ PGL(N+1)] and FMg ' [Hg,n/ PGL(N+ 1)]. Note that the theorem asserts that the quotient is independent of n.

Proof. We construct a functor p : FMg → [Hg,n/ PGL(N + 1)] which takes FMg to [Hg,n/ PGL(N + 1)] as follows: Given a family of stable curves C →π B, let E → B be the principal PGL(N + 1) bundle ⊗n 0 associated to the projective bundle P(π∗(ωC/B)). Let π : C ×B E → E be the pullback family. The pullback of this projective bundle to E is 0 trivial and is isomorphic to P(π∗(ωC×B E/E)), so there is a map E → Hg,n which is clearly PGL(N + 1) invariant. If C0 → C π0 ↓ π ↓ φ B0 → B 21 0 ∗ 0 0 is a morphism in FMg , then π∗(ωC /B ) ' φ π∗(ωC/B), so we obtain a morphism of associated PGL(N + 1) bundles E0 → E ↓ ↓ . B0 → B If C is a stable curve defined over an algebraically closed field k then any non-trivial automorphism of C induces an automorphism of 0 ⊗n the projectivized n-canonical linear system P(H (ωC )). This automorphism must be non-trivial because it acts non-trivially on the n- canonical embedding of C into PN . As a result our functor is faithful. Conversely, any non-trivial element φ ∈ PGL(N + 1) which leaves the n-canonical curve C,→ PN invariant must act non-trivially on C. The reason is that the fixed locus of φ is necessarily a proper linear subspace of PN . However, the n-canonical embedding of C can not be contained in a proper linear subspace. This implies that the functor p is full. Now if E → B is an object of [Hg,n/ PGL(N + 1)] then there is πE a family CE → E of curves of genus g together with an isomorphism N P(πE,∗(ωCE /E)) ' PE , where PGL(N + 1) acts by changing the polar- ization. The morphism CE → E has a PGL(N+1)-linearized relatively ample line bundle and the quotient B = E/ PGL(N + 1) is a scheme. Descent theory says that in this case a quotient C = CE/ PGL(N + 1) also exists as a scheme. Then CE ' C ×B E, so the object E → B is isomorphic to of p(C → B). Therefore p is an equivalence of categories.

Corollary 3.1. FMg and FMg are separated Deligne-Mumford stacks.

Proof. We have just shown that FMg and FMg are quotients of a scheme by a smooth group scheme. Moreover, every stable curve defined over an algebraically closed field has a finite and reduced automorphism group, so the stabilizers of the geometric points are finite and reduced. Therefore, they are Deligne-Mumford stacks by Corollary 2.2.

The separatedness of FMg and FMg follows from [DM, Theorem 1.11] which states that if C0/B and C/B are stable curves then the map 0 IsoB(C ,C) → B is ﬁnite.

3.2. Properness of FMg . Given that FMg is a Deligne-Mumford stack, the valuative criterion of properness and the following stable reduction theorem show that it is proper over Spec Z. Theorem 3.3. Let B be the the spectrum of a DVR with function ﬁeld K, and let X → B be a family of curves such that its restriction 22 0 XK → Spec K is a smooth curve. Then there is a ﬁnite extension K /K and a unique stable family X0 → B0 where B0 is the normalization of B in K0 such that the restriction X0 → Spec K0 is isomorphic to 0 XK ×K K .

Remark 3.3 (Remarks on the proof of Theorem 3.3). The properness of FMg follows from the observation: To check the properness of a morphism using the valuative criterion, it suffices to consider maps where the image of Spec K is contained in a fixed dense open substack (see the discussion after the statement of [DM, Theorem 4.19]). This theorem was originally proved (but not published) in characteristic zero by Mumford and Mayer ([GIT, Appendix D]). There is a relatively straightforward algorithmic version of this theorem in characteristic 0 which I learned from Joe Harris. Blow up the singular points of the special fiber of X/B until the total space of the family is smooth and the special fiber has only nodes as singularities. The modified special fiber will have a number of components with positive multiplicity coming from the exceptional divisors in the blowups. Next, do a base change of degree equal to the l.c.m. of the multiple components. After base change all components of the special fiber will have multiplicity 1. Then contract all (-1) and (-2) rational components in the total space. (That this can be done follows from the existence of minimal models for surfaces.) The special fiber is now stable. Furthermore, the total space of the new family is a minimal model for the surface. Since minimal models of surfaces are unique, the stable limit curve is unique. This algorithmic proof fails in characteristic p > 0, because after blowing up some components of the special fiber may have multiplicity divisible by p. In this case, it will not be possible to make the component become reduced after base change. Deligne and Mumford proved the stable reduction theorem in arbitrary characteristic using Neron models of the Jacobians of the curves ([DM]). Later Artin and Winters [AW] gave a direct geometric proof using the theory of curves on surfaces.

3.3. Irreducibility of FMg and FMg . Using the description of the moduli stacks as quotients of Hg and Hg we can deduce properties of the stacks from the corresponding properties of the Hilbert scheme. In particular, deformation theory shows that Hg and Hg are smooth over

Spec Z ([DM, Cor 1.7]). Since the map Hg → FMg (resp. Hg → FMg ) is smooth we see that FMg is smooth. 23 Further analysis [DM, Cor 1.9] shows that the scheme Hg − Hg rep- resenting polarized, singular, stable curves is a divisor with normal crossings in Hg. This property descends to the moduli stacks.

Theorem 3.4. [DM, Thm 5.2] FMg is smooth and proper over Spec Z.

The complement FMg −FMg is a divisor with normal crossings in FMg . The main result of [DM] is the following theorem:

Theorem 3.5. [DM] FMg has irreducible geometric ﬁbers over Spec Z. Remark 3.4. Deligne and Mumford gave two proofs of this theorem. In both cases they deduce the result from the classical characteristic 0 result stated below. We outline below their second proof, which uses Deligne-Mumford stacks.

Proposition 3.1. FMg ×Spec Z Spec C is irreducible.

Proof. It was shown classically that there is a space Hk,b parametrizing degree k covers of P1 simply branched over b points defined over the complex numbers. In [Fu], Fulton showed that the functor FHk,b whose sections over a base B are families of smooth curves C → B together 1 with a degree k map C → PB expressing each geometric fiber as a cover of P1 simply branched over b points is represented by a scheme which we also call Hk,b. In characteristic greater than k it is a finite 1 b étalecover of Pb = (P ) − ∆, where ∆ is the union of all diagonals (This fact was known classically over C.) Since Pb is smooth, Hk,b is also smooth (in sufficiently high characteristic). Thus, Hk,b is irreducible if and only if it is connected. Over Spec C, the connectedness of Hk,b in the classical topology was demon- strated by Hurwitz who showed that the monodromy group of the covering Hk,b → Pb acts transitively on the fiber over a base point in Pb for all k, b. Thus, Hk,b is irreducible in sufficiently high characteristic. Since there is a universal family of branched covers Ck,b → Hk,b there is a map Hk,b → FMg (where g = b/2 − k + 1). By the Riemann-Roch theorem for smooth curves, every curve of genus g can be expressed as a degree k cover of P1 with b simple branch points, as long as k > g + 1. Thus for k (and thus b) sufficiently large, the map is surjective. Therefore FMg is irreducible in characteristic greater than k, and thus FMg ×Z C is irreducible.

Proof. (Outline of the proof of Theorem 3.5) Since FMg − FMg is a divisor FMg is irreducible if and only if FMg is as well. The ﬁbres of

FMg → Spec Z are smooth, so it suffices to show that they are connected. The morphism FMg → Spec Z is proper, flat and has smooth 24 geometric fibers, so by Theorem 2.4 the number of connected components of the geometric fibers is constant. By the proposition the geometric fiber F × Spec is connected, so every geometric Mg Spec Z C fiber is connected. Remark 3.5. In [HM] Harris and Mumford constructed a compactification of Hk,b where the boundary represents stable curves expressed as branched covers of chains of P1’s. The existence of this compactification implies that every smooth curve admits degenerations to singular stable curves. Fulton [Fu82] used this fact to resurrect an argument of Severi giving a purely algebraic proof of the irreducibility of FMg in characteristic 0. This combined with the results of [DM] give a purely algebraic proof that FMg is irreducible in arbitrary characteristic.

4. Construction of the moduli scheme As we have previously seen, the moduli stack is a quotient stack of a smooth scheme Hg by PGL(N + 1). In this, the final section, we discuss the construction of a quotient scheme Hg/ PGL(N + 1) over an algebraically closed field k. We first prove that such a scheme is unique and is the coarse moduli space for the quotient stack. We then briefly discuss Gieseker’s GIT construction of a quotient scheme. 4.1. Moduli schemes and geometric quotients. This definition is completely analogous to Mumford’s definition ([GIT, p. 99]) of a coarse moduli scheme mentioned above. Definition 4.1. The moduli space of a Deligne-Mumford stack F is a scheme M together with a morphism π : F → M, such that (*) for any algebraically closed field k there is a bijection between the set of isomorphism classes of objects in the groupoid F (Ω) and M(Ω), where Ω = Spec k. Furthermore, M is universal in the sense that if N is a scheme then any morphism F → N factors through a morphism M → N. Remark 4.1. The universal property guarantees that the moduli scheme is unique if it exists. Though it is sufficient for our purpose, there are two drawbacks to this definition. (1) In characteristic p, the property of being a moduli space is not invariant under base change; i.e. if M 0 → M is a morphism then M 0 0 need not be the moduli space of M ×M F . As a result, Gillet [?] gave an alternative definition: Namely, M is a moduli space for F if the morphism F → M is proper and a bijection on geometric points. This notion is clearly preserved by base change but such a scheme is not 25 unique. However, two moduli spaces are universally homeomorphic as schemes. (2) If F is a Deligne-Mumford stack, then F may not have a moduli scheme (in either sense). If one is willing to look in the category of algebraic spaces a theorem of Keel and Mori [KM] states that, under very mild assumptions, if F is a D-M stack3 there is an algebraic space M such that M is a moduli space for F ; i.e. there is a morphism F → M which is a bijection on geometric points and which is universal for maps of algebraic spaces. However, these two notions may differ. For example, A1 is the moduli scheme (in our sense) of the non-separated algebraic space [X/ SL(2, C)] discussed in Example 2.13. Definition 4.2. [GIT, Definitions 0.5, 0.6] Let X/S be a scheme and let G/S be a smooth affine group scheme acting on X. An S-scheme Y is a geometric quotient of X by G if there is a morphism X → Y such that (1) f is G invariant. (2) The geometric fibers of f are orbits. (In particular f is surjective). (3) f is universally submersive, i.e. U ⊂ Y is open iff f −1(U) is open, and this property is preserved by base change. G (4) f∗(OX ) = OY . Remark 4.2. The purpose of the geometric invariant theory developed by Mumford is to construct geometric quotients for the action of a geometrically reductive group The definition of a geometrically reductive group is given in [GIT, Appendix A]. In characteristic 0 this notion is the same as the notion of linear reductivity; i.e. every representation decomposes as a direct sum of irreducibles. However, in characteristic p the only linear reductive groups are extensions of tori by finite groups of order prime to p. However, for the purpose of these notes, it suffices to know that SL(N + 1, k) is reductive for a field k. The following is a restatement of [GIT, Prop 0.1]. Proposition 4.1. A geometric quotient is a categorical quotient. That f g is, if X → Y is a geometric quotient and if X → Z is a G invariant morphism, then there is a unique morphism φ : Y → Z such that g = φ ◦ f. Note that the proposition implies that if a geometric quotient exists then it is unique.

3The theorem of Keel and Mori also applies to a certain class of Artin stacks. 26 Now let X be a scheme with a G action such that the stabilizers of geometric points are ﬁnite and reduced. We have seen that the groupoid [X/G] is a Deligne-Mumford stack.

Proposition 4.2. If f : X → Y is a geometric quotient of X by G then Y is the moduli space of the stack [X/G]. If in addition the action of G on X is proper then the morphism [X/G] → Y is proper.

Proof. If Ω = Spec K where K is algebraically closed, then Hom(Ω,Y ) is, by Condition (2) of the definition, the set of orbits of K-valued points of X. This is exactly set of isomorphism classes in [X/G](Ω). Therefore, condition (*) is satisfied. Next suppose that [X/G] → N is a morphism to a scheme. It is easy to see that that the induced morphism X → N is G-invariant. By the universal mapping property of the quotient, the morphism X → N factors through Y . Thus, the morphism [X/G] → N also factors through Y , so Y is the moduli scheme for [X/G] If the action is proper then [X/G] is separated, so the morphism [X/G] → Y is also separated. Then the the universal submersiveness of f : X → Y implies that the morphism [X/G] → Y satisfies the valuative criterion of properness. The proof is given in [Vi, Proof of Prop. 2.11]

4.2. Construction of quotients by geometric invariant theory. From now on we will assume that all schemes are deﬁned over an algebraically closed ﬁeld k. In this paragraph we discuss the geometric invariant theory neces- sary to construct Mg and and Mg as quotients of Hilbert schemes of n-canonically embedded (stable) curves. Our source is [Gi, Chapter 0]. For a full treatment of geometric invariant the classic reference is Mumford’s [GIT]. Let X ⊂ PN be a projective scheme, and let G be a reductive group acting on X via a representation G → GL(N + 1).

Definition 4.3. (1) A closed point x ∈ X is called semi-stable if there exists a non-constant G-invariant homogeneous polynomial F such that F (x) 6= 0. (2) x ∈ X is called stable if: dim o(x) = dim G (where o(x) denotes the orbit of x) and there exists a non-constant G-invariant polynomial such that F (x) 6= 0 and for every y0 in XF = {y ∈ X|F (y) 6= 0}, o(y0) is closed in XF . 27 Let Xss denote the semi-stable points of X, and Xs denote the stable points. Then Xs ⊂ Xss are both open in X. However, they may be empty. The following is the first main theorem of geometric invariant theory. Theorem 4.1. There exists a projective scheme Y and an affine, uni- ss versally submersive morphism fss : X → Y such that Y is a categorical quotient (such a morphism is often called a good quotient in the lit- erature). Furthermore, there exists U ⊂ Y open such that f −1(U) = Xs s s and fs : X → U is a geometric quotient of X by G. Remark 4.3. Proposition 4.2 implies that U is the moduli space of [Xs/G]. Moreover, geometric invariant theory also says that G acts properly on Xs so the morphism [Xs/G] → U is proper. 4.3. Criteria for stability. Let X ⊂ PN be a projective scheme, and let X˜ ⊂ AN+1 be the affine cone over X. Assume as above, that a reductive group G acts on X via a representation G → GL(N + 1). Then G acts on X˜ as well. The stability of x ∈ X can be rephrased in terms of the stability of the pointsx ˜ ∈ X˜ lying over x. Proposition 4.3. [GIT, Chapter 1, Proposition 2.2 and Appendix B] A geometric point x ∈ X is semi-stable if for one (and thus for all) x˜ ∈ X lying over X, 0 ∈/ o(˜x). The point x is stable if o(˜x) is closed in AN+1 and has dimension equal to the dimension of G. The second main theorem of geometric invariant theory is Mumford’s numerical criterion for stability which we now discuss. Definition 4.4. A 1-parameter subgroup of G is a homomorphism λ : Gm → G. This will be abbreviated to “λ is a 1-PS of G”. Now if λ is a 1-PS of G, then the since λ is 1-dimensional, there is a N+1 basis {e0, . . . , eN } of A such that the action of λ is diagonalizable ri with respect to this basis; i.e. λ(t)ei = t ei where t ∈ Gm and ri ∈ Z. P ˜ Ifx ˜ = xiei ∈ X, then the set of ri such that xi is non-zero is called the λ-weights ofx ˜. Note that if x ∈ PN then the λ-weights are the same for all points in AN+1 − 0 lying over x. Definition 4.5. x ∈ X is λ-semi-stable if for one (and thus for all) x˜ ∈ X˜ lying over x,x ˜ has a non-positive λ weight. A point x is λ-stable ifx ˜ has a negative λ-weight. Theorem 4.2. [GIT] A point x ∈ X is (semi)stable if and only if x is λ-(semi)stable for all 1-PS λ : Gm → G. 28 Remark 4.4 (Remark on the Proof). It is easy to see that if x is unstable (i.e. not semi-stable) with respect to λ : Gm → G then x is unstable. The reason is that if all the weights of λ are positive then 0 will be in the closure of the G-orbit ofx ˜ in AN+1 − 0. The converse is more difficult. Example 4.1. (cf. [GIT, Proposition 4.1]). The set of homogeneous forms of degree 4 in two variables forms a 5-dimensional vector space V . We will view P(V ) as the space parametrizing 4-tuples of (not necessarily) distinct points in P1. There is a natural action of SL(2) on V inducing an action on P(V ). Let us use the numerical criterion to determine the stable and semi-stable locus in P(V ). If v ∈ V is a form of degree 4 and λ is a 1-PS subgroup of SL(2), 4 3 2 2 3 4 then we can write v = a4X0 +a3X0 X1 +a2X0 X1 +a1X0X1 +a0X1 , and r −r λ acts by λ(t)(X0) = t X0, λ(t)(X1) = t X0 and r > 0(the weight on X1 must be the negative of the weight on X0, since λ maps to SL(2)). The possible weights of v are {4r, 2r, 0, −2r, −4r}. In order for v to be λ-stable one of a1 or a0 must be non-zero. It is λ-semi-stable if one of a2, a1 or a0 is non-zero. On the other hand, we can consider the 1-PS, −r r τ which acts by τ(t)X0 = t X0 and τ(t)X1 = t X1. In order for v to be τ-stable one of a4 and a3 must be non-zero, while it is τ-semi-stable if a2 is non-zero. Combining the conditions imposed by λ and τ we see that if v is stable, then one of a0 or a1 is non-zero and one of a3 or a4 is non-zero. This condition is equivalent to the condition that (1 : 0) and (0 : 1) are not multiple points of the subscheme of P1 cut out by the form v. Likewise, v is semi-stable if (1 : 0) or (0 : 1) is cut out with multiplicity no more than 2. Finally v is unstable if (1 : 0) or (0 : 1) is cut out with multiplicity more than 2. From this analysis it is clear that if v ∈ V cuts out 4 distinct points then it will be stable for every 1-PS. Likewise if v cuts out a subscheme of P1 with each point having multiplicity 2 or less then it is semi-stable for every 1-PS. Conversely, if v cuts a point of multiplicity 3 or more 3 1 then v = X0 (a0X0 + a1X1) for some choice of coordinates on P . Then v will have strictly positive weights for a 1-PS λ acting diagonally by r λ(t)X0 = t X0 for r > 0. N+1 4.4. Gieseker’s construction of Mg. Let HilbP (t) be the Hilbert scheme of curves in PN with Hilbert polynomial P (t). Grothendieck’s uniform m-lemma says that if X ⊂ PN is a curve with Hilbert polynomial P (t), then there exists m >> 0 (independent of X) such 0 N 0 that the restriction map H (P , OPN (m)) → H (X, OX (m)) is sur- 0 jective and dim H (X, OX (m)) = P (m). Taking the P (m)-th exterior m VP (m) 0 N power of φm we obtain a linear map V = H (P , OPN (m)) → 29 VP (m) 0 H (X, OX (m)) = k unique up to scalars; i.e., an element of P(V m). The corresponding point in P(V m) is called the m-th Hilbert point of X and is denote Hm(X). In this way we obtain a map N+1 m HilbP (t) → P(V ). For m sufficiently large this map is an embedding. Both SL(N +1) and PGL(N +1) act on P(V m) via the m-th exterior power representation of SL(N + 1) → GL(V m). Now the action of SL(N + 1) factors through the action of PGL(N + 1) (the stabilizer of SL(N + 1) at a geometric point is the group of N + 1 roots of unity) so we have the following proposition. Proposition 4.4. If X ⊂ P(V m) then X → Y is a geometric quotient by SL(N + 1) if and only if it is a geometric quotient by PGL(N + 1). Proof. If X → Y is a geometric quotient by SL(N + 1) then the geometric fibers are SL(N + 1) orbits. These orbits are the same as SL(N+1) PGL(N+1) the PGL(N + 1) orbits. Likewise, OX = OX . Thus, PGL(N+1) OY ' f∗OX . Finally if W and V are PGL(N + 1) invariant, they are also SL(N + 1) invariant. Thus if they are disjoint, then since X → Y is an SL(N + 1) quotient, their images will be disjoint as well. Hence X → Y is a PGL(N + 1) quotient. The converse is similar. Let g ≥ 3 and d ≥ 20(g−1) be integers. Consider the Hilbert scheme N+1 N=d−g HP (t) of curves in P with Hilbert polynomial P (t) = dt − g + 1 (the curves parametrized necessarily have arithmetic genus g). The first step in Gieseker’s construction is to prove the following theorem. The proof is 10 pages long and uses the numerical criterion.

Theorem 4.3. [Gi, Theorem 1.0.0] There exists an integer m0 >> 0 such that if X is smooth then Hm0 (X) is SL(N + 1) stable.

Remark 4.5. The theorem is not necessarily true for arbitrary m0 >> 0. However there are inﬁnitely many m0 for which the theorem is true ([Gi, Remark after Theorem 1.0.0]). The next, and technically most diﬃcult step is to prove the following theorem. The proof takes 50 pages!

Theorem 4.4. [Gi, Theorem 1.0.1] For the same integer m0, every N+1 m0 ss point in HilbP (t) ∩ P(V ) parametrizes a Deligne-Mumford semi- stable curve.

N+1 Let U ⊂ HilbP (t) be the subscheme of semi-stable curves with re- N spect to the m0-th Hilbert embedding. Let ZU ⊂ PU be the restriction of the universal family of projective curves. As before, view a point 30 h ∈ U as parametrizing a curve Xh and a very ample line bundle L of degree d on X . Set U = {h ∈ U|L ' ωn }. This is a lo- h h c h Xh cally closed subscheme of U which is empty unless 2g − 2 divides d. Gieseker then proves that Uc is in fact closed in U. He also proves that Uc is smooth ([Gi, Theorem 2.0.1]) and parametrizes only all Deligne- Mumford stable curves; thus, Uc ' Hg,n. Since Uc is closed in U there is a projective quotient Uc/ SL(N + 1). Finally note that PGL(N + 1) (and thus SL(N + 1)) acts with finite stabilizers on points of Uc because the curves parametrized have finite automorphism groups. Hence the points of Uc are in fact SL(N + 1) stable. Thus a geometric quotient Uc/ SL(N + 1) exists. Since this is isomorphic to a geometric quotient Uc/ PGL(N + 1) ' Hg,n/ PGL(N + 1) we have succeeded in constructing a coarse moduli scheme for the stack of stable curves. References [AW] M. Artin, G. Winters, Degenerate fibres and stable reduction of curves, Topol- ogy, 10, 373-383 (1971). [DM] P. Deligne, D. Mumford, Irreducibility of the space of curves of given genus, Publ. Math. IHES, 36, 75-110 (1969). [Di] S. Diaz, A bound on the dimensions of complete subvarieties of Mg, Duke Math. J. 51 405-408 (1984). [Do] I. Dolgachev, Rationality of fields of invariants, in Algebraic geometry, Bow- doin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46 Part 2, 3-15 (1987). [EGA4] A. Grothendieck and J. Dieudonné, Eléments´ de géométrie algébrique (Etude´ locale des schémas et des morphismes des schémas), Publ. Math. I.H.E.S, 32 (1967). [Fu] W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. Math. 90 542-575 (1969). [Fu82] W. Fulton, On the irreducibility of the moduli space of curves, appendix to a paper of Harris and Mumford, Invent. Math. 67 87-88 (1982). [Gi] D. Gieseker, Tata lectures on moduli of curves, Springer Verlag, NY (1982). [?, Gil] H. Gillet, Intersection theory on algebraic stacks and Q-varieties, J. Pure Appl. Alg. 34 (1984), 193-240. [HM] J. Harris, D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 23-86 (1982). [KM] S. Keel, S. Mori, Quotients by groupoids, Ann. Math. 145 (1997), 193-213. [L-MB] G. Laumon, L. Moret-Bailly, Champs algébriques, book, to appear. [GIT] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Third Enlarged Edition, Springer Verlag, NY (1994). [Vi] A. Vistoli, Intersection theory on algebraic stacks and their moduli spaces, Invent. Math. 97, 613-670 (1989).

Department of Mathematics, University of Missouri, Columbia MO 65211 E-mail address: [email protected] 31